A210803 Triangle of coefficients of polynomials u(n,x) jointly generated with A210804; see the Formula section.

1, 1, 1, 1, 3, 2, 1, 8, 10, 3, 1, 22, 37, 21, 5, 1, 63, 125, 100, 45, 8, 1, 185, 409, 410, 260, 88, 13, 1, 550, 1321, 1562, 1240, 598, 169, 21, 1, 1644, 4238, 5706, 5331, 3258, 1319, 315, 34, 1, 4925, 13534, 20284, 21507, 15651, 8071, 2776, 578, 55, 1

Offset: 1

Author

Clark Kimberling, Mar 27 2012

Keywords

• nonn
• tabl

Comments

Row n starts with 1 and ends with F(n), where F=A000045 (Fibonacci numbers).

Column 1: 1,1,1,1,1,1,1,1,1,1,1,...

Column 2: A047849

Row sums: A003462

Alternating row sums: 1,0,0,0,0,0,0,0,0,...

For a discussion and guide to related arrays, see A208510.

Essentially the same triangle as (1, 0, 3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Jul 11 2012

Examples

			First five rows:
1
1...1
1...3....2
1...8....10...3
1...22...37...21...5
First three polynomials u(n,x): 1, 1 + x, 1 + 3x + 2x^2.
		

Crossrefs

Cf. A210804, A208510.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + j)*v[n - 1, x] + c;
d[x_] := h + x; e[x_] := p + x;
v[n_, x_] := d[x]*u[n - 1, x] + e[x]*v[n - 1, x] + f;
j = 0; c = 0; h = -1; p = 3; f = 0;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A210803 *)
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A210804 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]   (* A047849 *)
Table[v[n, x] /. x -> 1, {n, 1, z}]   (* A000302 *)
Table[u[n, x] /. x -> -1, {n, 1, z}]  (* A000007 *)
Table[v[n, x] /. x -> -1, {n, 1, z}]  (* A000007 *)

Formula

u(n,x)=u(n-1,x)+x*v(n-1,x)+1,

v(n,x)=(x-1)*u(n-1,x)+(x+3)*v(n-1,x),

where u(1,x)=1, v(1,x)=1.

T(n,k) = 4*T(n-1,k) + T(n-1,k-1) - 3*T(n-2,k) - 2*T(n-2,k-1) + T(n-2,k-2), T(1,0) = T(2,0) = T(2,1) = T(3,0) = 1, T(3,1) = 3, T(3,2) = 2, T(n,k) = 0 if k<0 or if k >= n. - Philippe Deléham, Jul 11 2012

G.f.: (-1+3*x)*x*y/(-1+4*x-3*x^2-2*x^2*y+x*y+x^2*y^2). - R. J. Mathar, Aug 12 2015